IRB Asset and Default Correlation: Rationale for the Macroprudential Add-ons to the Risk-Weights - H. Penikas
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IRB Asset and Default Correlation: Rationale for the Macroprudential Add-ons to the Risk-Weights WORKING PAPER SERIES No. 56 / July 2020 H. Penikas
IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
ЯНВАРЬ 2019
Henry Penikas
Bank of Russia, Research and Forecasting Department
Email: henry_penikas@cbr.ru
Bank of Russia Working Paper Series is anonymously refereed by members of the Bank of Russia Re-
search Advisory Board and external reviewers.
Author is grateful for the discussions at the research workshops held in December, 2019 within the Cen-
tral Economics and Mathematics Institute and the Bank of Russia.
Cover image: Shutterstock.com
© Central Bank of the Russian Federation, 2020
Address: 12 Neglinnaya street, Moscow, 107016
Tel.: +7 495 771-91-00, +7 495 621-64-65 (fax)
Website: www.cbr.ru
All rights reserved. The views expressed in this paper are solely those of the authors and do not necessarily reflect
the official position of the Bank of Russia. The Bank of Russia assumes no responsibility for the contents of the pa-
per. Any reproduction of these materials is permitted only with the express consent of the authors.
2IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
ABSTRACT
Basel III allows for the use of statistical models. It is called the internal-ratings-based
(IRB) approach and is based on the (Vasicek, 2002) model. It assumes assets returns are standard
normally distributed. It suggests incorporating different asset correlation (R) functions to assess
credit risk for the loan portfolio, or the risk-weighted assets (RWA). The asset correlation func-
tion solely depends on the individual default probability (PD) given certain credit exposure type.
At the same time, the IRB approach requires developing PD models to predict the dis-
crete default event occurrence. This means that the IRB approach is based on the Bernoulli trials.
We investigate the impact of the asset returns’ correlation for the Bernoulli trials. We show that
when Bernoulli trials are considered, the credit risk estimation significantly deviate from the val-
ues derived under the normality assumption of asset returns.
We investigate the simulated and real-world credit rating agencies’ data to specifically
demonstrate the scale of the credit risk underestimation by the IRB approach. Therefore,
macroprudential add-ons are of use to offset such IRB limitations.
Keywords: Basel II, IRB, correlated defaults, asset correlation, binomial distribution, Bernoulli
trials, macroprudential add-ons.
JEL codes: C25, G21, G28, G32, G33.
3IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
1. INTRODUCTION
Capital adequacy ratio (CAR) became an internationally accepted indicator of the bank's
financial standing in 1988 when the Basel I Accord was adopted by the Basel Committee on
Banking Supervision (BCBS, 1988). CAR equals to the amount of the bank own funds (capital)
divided by the amount of the risk-weighted assets (RWA). RWA is a product of a risk-weight
(RW) and an asset (A) amount. Originally Basel I defined several categories of fixed risk-
weights.
In 2006 Basel II allowed using bank own loan default statistics and mathematical models
to estimate the risk-weights by banks themselves (BCBS, 2006). The approach was called an in-
ternal-ratings-based (IRB) one. By 2019 around two thousand of the world largest banks operat-
ed IRB (Ermolova & Penikas, 2019, p. 106). We recommend referring to (Penikas, 2015) for
more details on the overall banking regulation evolution developed by the Basel Committee and
to (Penikas, 2020a) for more details on the evolution of the IRB approach.
Concisely, the IRB approach is based on (Vasicek, 2002) loan portfolio model. It departs
from the key assumption of the normally distributed asset returns. However, the Basel Commit-
tee requires predicting the discrete default events within the IRB approach. This means that there
is a change in (Vasicek, 2002) key model assumption. That is why it is important to investigate
how the transit from the Gaussian asset returns to the Bernoulli random variables standing for
default events may change the credit risk assessment. In case the deviations are material, there
comes a need to offset it in the day-to-day supervisory practice of the regulator. Macroprudential
add-ons (mark-up) to the IRB risk-weights are of value here. Thus, the objective of the current
working paper is to determine the values and drivers for such macroprudential add-ons given the
difference in IRB assumptions from the banking practice.
There are two primarily findings. First, we show that the values of the discrete default
events’ correlation varies because of the DR variances given the same PD. The IRB approach
does not account for the DR variance as a driver when computing the asset correlations. Second,
the offered parametrization for the default correlation enables to recalibrate the AC function with
the recent data available to the credit institution, i.e. to have both IRB inputs up to date (PD and
AC). Such an update may signal whether the regulator has to introduce non-zero
macroprudential add-ons to the risk-weighted assets or not.
To guide the reader on how we got the AC and DC function parametrization and the val-
ues of the macroprudential add-ons, the paper has the following structure. Section 2 presents the
literature overview. It consists of two parts. Part 1 briefly shows the Basel IRB framework that
takes the asset correlation as an input. Asset and default correlation definitions are discussed
4IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
there. Part 2 discusses how various researchers handled the issue of the asset and default correla-
tion evaluation assuming the Gaussian asset returns. Part 3 briefly refers to macroprudential pol-
icy tools. Those are used to compensate for the shortcomings of the microprudential regulation,
including the IRB one. Section 3 proves the suggested parametrization of the asset and default
correlation for the discrete default events. Section 4 runs the robustness check for the suggested
parametrization using simulated data with negative, neutral and positive correlation of discrete
defaults. Section 5 presents evidence from the publicly available data to demonstrate the impact
of the suggested parametrization. Particularly, we were able to find out the situations when the
IRB approach and 2003-04 AC function assuming correlated the Gaussian asset returns ca. twice
underestimate the credit risk compared to the approach assuming the correlated discrete defaults.
Section 6 concludes underlining the importance of such an underestimation treatment via
macroprudential add-ons to the IRB risk-weights.
2. LITERATURE REVIEW
PART 1. BASEL IRB FRAMEWORK
(Vasicek, 2002) model underlies the IRB approach in Basel II and III accords (BCBS, 2005c).
(Merton, 1974) model forms the basis for the Vasicek one. Thus, Merton defined default for the
company as a situation when the market value of its publicly traded assets goes below the
amount of its debt. Vasicek used Merton definition and expanded it by saying that the market
value of the i-th company asset, or its asset (value) return, ( ) depends on the two factors: sys-
temic ( ) and idiosyncratic ( ) as follows in (1).
(1) ,
where - asset (value) return of the i-th borrower, - systemic factor,
- idiosyncratic factor, - dependence parameter, - standard normal distri-
bution.
(Vasicek, 2002) assumes that the systemic factor contributes to the asset market value
with a share of square root of . As a note for future discussion, he does neither introduce de-
fault correlation, nor suggests any parametrization for R. However, he notes, if there was no ,
the resultant default rate distribution would converge to a Gaussian one.
Terms ‘asset and default correlations’ appear in the working paper No. 14 (BCBS,
2005). Specifically, R is called asset correlation. We will cover the ‘default correlation’ defini-
tion from (BCBS, 2005) later in Part 2 of the literature review. However, the working paper
(BCBS, 2005) has a different expression for the asset value that is conventional to the proba-
bility theory:
(2) .
5IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
Formula (2) is conventional to the generation of a normally distributed random variable
Z that co-depends with another normally distributed random variable X with the correlation coef-
ficient ( ). Formula (1) is related to formula (2) via the following relationship:
(3) .
Such a fact does not imply that the asset or default correlation becomes neither signifi-
cant, nor insignificant. As we show later in (23), the higher the variance of the default rate is, the
higher the default correlation becomes. As the Figure 3 below shows, the closer the default
rate gets to zero or one, the larger the asset correlation gets. Thus stands for the mathemati-
cal correlation (let us refer to it as default correlation, DC, as (Foulcher, Gouriéroux, & Tiomo,
2005) also do), whereas R stands for the IRB asset correlation (AC). By introducing such a defi-
nition, asset correlation R is always positive. However, the default correlation might still be neg-
ative. From the credit risk management perspective formula (1) does not allow to account for the
risk diversification when there is a negative correlation of the credit risk events. Formula (2)
does allow for the risk diversification, as we show below, specifically in case of a finite and not
large number (ca. one thousand) of observations (the total number of borrowers or credit facili-
ties). (Gordy M. , 2000, pp. 147, eq. C.1) and (Gordy & Heitfield, 2010, pp. 46, eq. 3) also use
formula (2) allowing for negative correlation coefficient ( ).
However, BCBS decides to proceed with the formula (1). Formula (1) serves the base-
line for the following ultimate IRB risk-weighting formula (BCBS, 2005c) of Basel II and then
Basel III. Without loss of generality, we demonstrate formula (4) without subtraction of the ex-
pected loss part (-PD) and without multiplication by loss given default (LGD) and exposure at
default (EAD). in the presented form is the total capital requirement for the credit risk. Reg-
ulator requires decomposing it in two parts: expected loss (EL) and unexpected loss (UL). EL
goes into the numerator of the CAR formula; UL – to CAR denominator. Later we will show
policy implications with respect to the total capital requirement. That is why it is sufficient to
proceed with formula (4). It assumes that there is an infinite number of infinitely small identical
exposures, i.e. the portfolio is perfectly granular.
(4) ,
where – capital requirement for i-th borrower or facility; acts as a multiplier for assets
within the CAR denominator;
0.999 - confidence level for systemic risk-factor (X) realization;
– standard normal probability distribution function; – its inverse;
PD - probability of default lying in the range of [0; 1], its stands for the idiosyncratic risk-factor
value.
6IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
The AC function reflects the degree of borrowers’ co- or counter-dependence. We may
deem it to be a regulatory novelty as Vasicek did not suggest any functional dependence of R
and PD. This is why our further modification of does not make it impossible to apply the gen-
eral Vasicek model. The IRB AC function is as follows (see par. 53, 54, 64, 118, 120, pp. 63-75
of (BCBS, 2017)).
(5) ,
where - weight as a function of PD and calibration coefficient ( ) in the follow-
ing form
(6) ,
where R_... – the asset correlation; – the fixed value if R is independent of PD; – mini-
mum value of the asset correlation attained at the largest PD; – maximum value of asset
correlation attained at the smallest PD. Please, refer to Annex 1 in (Penikas, 2020a) for the pa-
rameter values per the credit exposure types.
As we can see from (5), R depends only on PD. BCBS wanted to reflect the following
idea. The higher PD is, the lower the R value is. This means, the less creditworthy the borrower
is, the less its credit risk is augmented when crisis comes in all other things being equal. The rea-
son for this is that one’s risk assessment is already relatively high. Figure 1 has the visual repre-
sentation of various R functions.
30
25
Corporate
Asset Correlation (R), %
20 SME
15 HVCRE
QRR
10
Mortgage
5 Other retail
SIFI
0
0% 20% 40% 60% 80% 100%
Probability of Default (PD)
Figure 1. Asset correlation per credit product types according to Basel II (III).
7IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
Basel working paper 14 (BCBS, 2005, p. 51) postulates that R values were calibrated in
2003 using quantitative impact study (QIS) 3. 1 This raises an issue whether the preserved R
functions' calibration is representative in 2020 and, if it is not, then how we should recalibrate it;
or introduce the macroprudential add-ons to the risk-weights to adjust for the material discrepan-
cies that may come out.
The key implication of the AC functions variation by product types is the fact that capi-
tal requirements differ given the same default statistics. Annex 2 offers two examples (see row
7): for mean DR equal to 5% and 10%. For instance, for the mean DR of 5% and DR variance of
1% the capital requirement ( ) – i.e. the proxy for the mean DR and a bonus for asset (de-
fault) correlation – varies from the lowest of 14.73% of the credit exposure amount if it is quali-
fying revolving retail (QRR) loan to 33.11% if it is a loan to a SIFI. We may compare both fig-
ures as they are ultimately summed up when computing total RWA.
To sum up, the IRB framework does not allow for the negative default dependencies.
The asset correlation function’s parametrization may be outdated and may require validation and
recalibration or the macroprudential add-ons to the risk-weights introduction.
PART 2. APPROACHES TO ASSET AND DEFAULT CORRELATION MODELING
There are several milestones in the literature when we trace asset and default correlation
modeling. To list those chronologically, they are (Vasicek, 2002), (Duffie & Singleton, 1999),
(Li D. , 2000), (Nagpal & Bahar, 2001), (Lopez, 2002), (Duffie & Singleton, 2003), (BCBS,
2005), (Blochwitz, Martin, & Wehn, 2006), (Patel & Pereira, 2008), (Ozdemir & Miu, 2009),
(Duellmann, Kull, & Kunisch, 2010), (Vozzella & Gabbi, 2010), (Li, Shang, & Su, 2015),
(Khorasgani & Gupta, 2017), (Wunderer, 2019), (Baldwin, Alhalboni, & Helmi, 2019).
(Duffie & Singleton, 1999) are the first authors to mention the correlation of defaults as
a formal term. They discuss the importance of such a feature for the credit portfolio management
by showing the increased DR variance in its presence. However, they do not attempt calibrating
it to historical data. Later (Li D. , 2000) and (Duffie & Singleton, 2003) discuss how copulas
might be applicable for the joint default dependence simulation. Though both use the Gaussian
copula, (Li D. , 2000) merely shows the methodology of how to apply it, whereas (Duffie &
Singleton, 2003) claim that the Gaussian copula is inadequate in fitting the data.
(BCBS, 2005, p. 51) suggests modification of a PD model accuracy validation proce-
dure (i.e. the binomial test) given the presence of the asset correlation. It still warns that direct
use of recommended asset values from QIS 3 (please, see Annex 1 in (Penikas, 2020a, p. 98))
would significantly extend the confidence interval for the binomial test. This implies making
1
BCBS distributed QIS 3 data collection template in November 2002 and collected it in 2003.
URL: https://www.bis.org/bcbs/qis/qis3.htm [accessed: December 13, 2019].
8IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
conclusions that the model is accurate more often than it is really needed or expected by some-
one. As a sort of response to (BCBS, 2005), (Blochwitz, Martin, & Wehn, 2006, p. 295) suggest
using default, not asset correlation in the Basel IRB risk weighting formula. (Blochwitz, Martin,
& Wehn, 2006, p. 295) also note that in practice default correlation may be up to 3%. This coin-
cides with the findings for the French data estimates by (Foulcher, Gouriéroux, & Tiomo, 2005)
and for the German companies by (Hamerle & Rosch, 2006, p. 21). Both values are much lower
than the values from the Basel AC function resulting in a compromise, i.e. the confidence inter-
val for the portfolio PD forecast becomes larger than in the case of the default independence,
though smaller than if validators used Basel II AC values. As a result, validators deem models to
be accurate less often than if when they used the IRB asset correlation R values.
(Lopez, 2002) and (Patel & Pereira, 2008) try to investigate the drivers of asset and de-
fault correlation. (Lopez, 2002) runs a more descriptive analysis, whereas (Patel & Pereira, 2008)
undertake a regression one. However, (Lopez, 2002) obtains asset correlation values that are
comparable to the regulatory one. (Patel & Pereira, 2008) use a proxy for default correlation
from the copula dependence parameters. The latter ones are computed for the stock asset returns
during five years of data. Thus, the major shortcoming of (Patel & Pereira, 2008) work is that
they do neither demonstrate the distribution of the default rates, nor the derived default correla-
tions, nor their proxies, nor run a goodness-of-forecast testing for the obtained proxy default cor-
relation values.
Let us review the approach to the default correlation (DC) definition more closely.
(Nagpal & Bahar, 2001) offer DC parametrization of the type for two credit risk categories (rat-
ings) i and j:
(7) ,
where is the joint default probability for the two identical random variables (variates)
and of a following type:
(8)
where is the default event occurrence for a single borrower (facility) when the asset
(value) is less than the interest rate on its debt, i.e. ; is the marginal (individual) de-
fault probability.
(BCBS, 2005, p. 48) offers a default correlation (DC) formula assuming the correlated
Gaussian asset returns. Author was unable to find the publicly available first version of the men-
tioned guidance if not to consider a page within (BCBS, 2000, pp. 38-39). (Aramonte & Avalos,
9IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
2020) seem to also use such an approach though they do not disclose granular approach to com-
putation.
(9) ,
where is a bivariate standard normal cumulative density function for the two random
variates and , such that or , is the in-
verse standard normal probability distribution function; and asset correlation is
. Illustrative values of are available in Annex 4. Default correlation may
be expected to be low when one considers underlying Gaussian asset value returns (Fantazzini,
2009, p. 119). However, even for Gaussian random variates the shadow area from Figure 2 in
(Fantazzini, 2009, p. 119) strongly depends upon the chosen threshold (i.e. debt levels from
the Vasicek model).
When one compares (7) to (9), one may conclude that (BCBS, 2005, p. 48) assumes the
following (Gordy M. , 2000, pp. 148, eq. C.4):
(10) .
Then resembles the concept of the upper (lower) tail dependence index for the
joint probability distribution (see (Nelsen, 1999), for instance). We wish to highlight once again
that such a transition in (10) is justified only for correlated Gaussian asset returns ( (Foulcher,
Gouriéroux, & Tiomo, 2005, pp. 5, formulas (2.5), (2.6)), (Gordy & Heitfield, 2010, p. 47),
(Wunderer, 2019)).
For the future discussion, let us remember the modification coming from the (BCBS,
2005, p. 48) formula (9) using (Vasicek, 2002) assumptions:
(11) .
(Duellmann, Kull, & Kunisch, 2010) find that asset correlation values differ when one
derives it either from the asset price (quote) data or from the default time series. (Wunderer,
2019, pp. 4, eq. 2.4) tries to explain this observation. Though (Wunderer, 2019) starts with the
discrete defaults, he uses (10) for the Gaussian random variates. (Wunderer, 2019) supposes that
the portfolio inhomogeneity is the cause of the differences in the correlation estimates based on
the asset return and the default statistics. To prove it, he departs from the following conventional
theoretical Bernoulli random variate variance definition:
(12)
and DR definition:
(13)
10IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
to come to the following DR variance as first shown in (Gordy & Heitfield, 2010, pp. 58, eq. 20):
(14)
He does not write this, but if we let n go to infinity, we obtain exactly the same component that
(BCBS, 2005) presented, i.e.
(15) .
If one joins formula (11) and (15), then one can obtain that
(16) .
As we show later, formula (16) is also the needed parametrization for the correlation of
discrete defaults (not Gaussian asset returns) when the number of borrowers goes to infinity.
To sum up, we have shown that the existing literature already provides the default cor-
relation parametrization for the discrete default events in (16). However, while (BCBS, 2019)
requires predicting discrete defaults, it still uses an assumption of the correlated Gaussian asset
returns in the Vasicek set-up of the IRB approach. The working paper by (Khorasgani & Gupta,
2017) also stated that the IRB asset correlation was wrongly parametrized.
As a robustness check for the formula (16), we derive it separately and model several
distributions with the various marginal default probabilities and the various default correlation
parameters. Then we compare the suggested parametrization with the parameter estimates when
the data generation process is known. As an alternative, we benchmark our results to that of
(Wunderer, 2019). We show that (Wunderer, 2019) overestimates asset correlation ( ) and capi-
tal requirements for more correlated and more risky borrowers if those related to the discrete de-
fault events (not to the continuous Gaussian asset returns).
PART 3. MACROPRUDENTIAL POLICY MEASURES.
We additionally investigate the real-world credit rating agencies’ data to demonstrate
the scale of the credit risk deviation. In cases where there is a substantial underestimation ac-
cording to the IRB approach, the macroprudential measures are useful to offset for those. The
macroprudential tools were discussed since 1970s in the auspices of the BCBS (Clement, 2010)
and since 2000 publicly (Crockett, 2000), (Borio, 2003). The extensive reviews of the
macroprudential policies are available in (Schoenmaker, 2014), (Kahou & Lehar, 2017), (Lubis,
Alexiou, & Nellis, 2019). (Penikas, 2020a) claims that the macroprudential add-ons to the risk-
weights (or increasing the risk-weights) should be preferred to mere change in the overall mini-
mal capital adequacy ratio (CAR) requirement from the bank stakeholders’ perspective. The lat-
ter change in the CAR minimum impacts only those banks that are close to the threshold. There-
fore, such a change may trigger no information signal (CAR value does not change) for the
11IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
banks with the different risk profiles, but that end up above the new CAR threshold. However,
the former macroprudential measure of a change in the risk-weights creates signals for all the
banks as it implies change in the CAR. Such a change is more pronounced for the banks that are
more exposed to the lending segment targeted by the macroprudential policy.
Thus, the macroprudential policy tools including add-ons to risk-weights form an inte-
gral part of the modern prudential banking regulation and supervision. Their efficiency is regu-
larly discussed and assessed. For instance, (BIS, 2020) reviews such tools implementation in the
Asian countries (Australia, Indonesia, New Zealand, Philippines, Thailand). Latin American
countries (Argentina, Brazil, Colombia, Mexico and Peru) are considered by (Gambacorta &
Murcia, 2020); Turkey is handled by ( arba & G ner, 2020); Southern Korea is the focus of
(Kim & Oh, 2020); European Union-wide effects are surveyed by (Budnik & Jasova, 2018) and
(Dautović, 2019); the Great Britain and Ireland are investigated by (Basten, 2020) and (McCann
& O'Toole, 2019). Macroprudential policy measures application for Russia is covered by
(Danilova & Morozov, 2017), (Ivanova, Andreev, Sinyakov, & Shevchuk, 2019), (Khotulev &
Styrin, 2020).
3. THEORETICAL PART
Our Assumptions
Here below we list the assumptions of the further research:
(1) We investigate solely the impact produced by a single parameter – the probability of de-
fault (PD). We do not consider the impact of other credit risk drivers, including LGD,
EAD, M (maturity). Such an investigation may be a focus of a separate paper. By doing
this, we follow (Vasicek, 2002) as he also assumed “gross loss (before recoveries)”.
Though we investigate PD and DR only, this working paper findings are applicable to both
the Foundation and Advanced IRB approaches as both depend upon a PD.
(2) We deal with a single PD model segment. If there is a single PD model for a bank portfolio
(e.g. for a narrowly focused business segment or a monoliner bank), than the working pa-
per findings are applicable to the entire portfolio.
(3) This model segment has a uniform distribution of borrowers by PD.
(4) Basel IRB approach requests methodologically differentiating approaches to PD model de-
velopment for corporate and retail claims. For the corporate claims a PD model should be
build at the borrower level. For the retail claims it should be built at the facility level. Dis-
regarding the differences in such a methodological treatment of the PD modeling, ultimate-
ly one sums up the risk-weighted assets for the corporate and retail borrowers. This means
that our research may compare results for the different asset classes.
12IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
(5) We assume that the PD used for the default correlation derivation reflects the long-run de-
fault experience of a bank for the lending segment of interest.
Default Correlation Parametrization Derivation
Consider the variance of the two identical Bernoulli random variates and introduced in (8)
(Kelbert & Sukhov, 2010, p. 78) or (Van Der Geest, 2005, p. 145):
(17) .
If is a correlation coefficient between identical and and , then their
covariance equals to:
(18)
We will call a default correlation as it estimates the correlation between Bernoulli trials that
indicate default. The variance of a sum of Bernoulli random variates equals to:
(19)
where for a single Bernoulli random variate , the variance is mentioned in formula (12).
Putting (12) and (18) into (19) yields us:
(20)
In case of the Bernoulli trials’ independence we simply have
(Kelbert & Sukhov, 2010, p. 83). (Witt, 2014, p. 4268) departs from the same standing
point to derive the formula for the distribution of the correlated Bernoulli trials, or the absolute
number of defaults. However, he does not provide an explicit parametrization for the default cor-
relation function for the case of the discrete default events. As a consequence, he does not evalu-
ate impact on the capital requirements. We do this in section 5 of the current working paper.
Consider the default rate (DR) as in formula (13). To estimate the credit risk at the port-
folio level, we substitute PD with its sample estimate in (20), i.e. with the mean value of DR
( ). Then the DR variance is:
(21)
From here we may derive the correlation value floor, that was noted by (Preisser & Qaqish,
2014):
(22) or for .
This means that when goes to infinity, one cannot expect a negative default correlation as a
single parameter over a credit portfolio. However, even for a thousand borrowers a minor nega-
tive correlation is feasible. Let us derive the default correlation value ( ) from (21). We have:
13IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
(23)
When one wishes to obtain asset correlation , it is sufficient to take the power of two for as
done in (3). Jointly formulas (3) and (23) for the correlated discrete defaults substitute formulas
(5), (6) for the correlated continuous Gaussian asset returns. If we simplify (23) by taking the
limit for going to infinity, then the proxy for the default correlation is:
(24)
In formula (24) we obtain the result of formula (16) that is a combination of (BCBS,
2005) and (Wunderer, 2019). It is interesting to mention here the survey by (Wehrspohn, 2004).
He argues that PD contributes more to the credit loss amount than the correlation. His statement
may be correct for the correlated Gaussian asset returns similarly to (Fantazzini, 2009, p. 119).
However, for the correlated Bernoulli trials both PD and r multiplicatively contribute to the de-
fault rate variance. The contribution depends upon their values.
For the regulatory purposes, we suggest the following modification to the Basel IRB risk-
weighting formula instead of formula (4). It allows for the negative default correlation.
(25)
where comes from formula (23).
There may be counterarguments against allowing for the negative default correlation in
(25). For instance, one may recall the lower boundary for the correlation value from (22), i.e. for
the large number of borrowers the correlation is non-negative. We need to underline here that
the presence of the negative default correlation is feasible in the case of low default portfolios
(LDP), i.e. when the number of borrowers is limited (BCBS, 2005b), (Benjamin, Cathcart, &
Ryan, 2006). As an illustration, one may imagine an IRB segment of lending to one hundred
large borrowers. Then a situation with is feasible. For comparison, (Penikas H. ,
2020b, p. 124) separately discusses that the LDP in itself may be an outcome of +100% default
correlation.
Let us now proceed with the robustness check of the proposed parametrization for the
default correlation .
4. ROBUSTNESS CHECK: SIMULATED DATA
To obtain the correlated Bernoulli random variate probability density distributions, we
use a genetic algorithm described by (Kruppa, Lepenies, & Jung, 2018). There are alternative
approaches for the correlated Bernoulli random variates’ distribution generation. Those include
(Lunn & Davies, 1998), (Zaigraev & Kaniovski, 2013), (Preisser & Qaqish, 2014), (Witt, 2014),
14IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
(Bakbergenuly, Kulinskaya, & Morgenthaler, 2016), (Chiu, Jackson, & Kreinin, 2017). (Kruppa,
Lepenies, & Jung, 2018) allow getting the distributions of the negatively correlated Bernoulli
outcomes. So we proceed with it.
(Kruppa, Lepenies, & Jung, 2018) algorithm has two steps. First, we generate a vector
with the required marginals, i.e. with the needed number of zeros (non-defaults) and ones (de-
faults) per columns. A column stand for the hypothetical borrower default status at the different
hypothetical points in time (rows). For simplicity we take ten variates ( ) and the number
of rows (observations) equals to 100. Second, one starts randomly perturbing (exchanging plac-
es) within any column so that marginals (mean PDs) do not change, but the default correlation
changes. We introduce an error as the square root of the sum of the element-wise differences of
the two correlation matrices: the one under perturbation and the target one. The target correlation
matrix has r values everywhere except the principal diagonal. For the simplicity, we consider the
matrix differences only for its half, i.e. for the below diagonal elements only. The optimization
procedure is iterative and may not always converge, as (Kruppa, Lepenies, & Jung, 2018) note.
That is why we set limits on the acceptable error (10%) and the maximum number of iterations
(10 thousand). We report both in Annex 1 for the found solution (see columns 7 and 6, respec-
tively).
For the simulation purpose, we choose four values of the marginal PD (5, 10, 25, 50%)
and six values of the default correlation r (-10, -5, 0, +5, +10, +20). For each parameter set we
generate a new initial DR distribution. That is why there may be statistically insignificant differ-
ences in the mean DR values given the same PD input in Annex 1 (compare columns 8 and 3,
respectively).
One may notice from Figure 2, that for the significantly positive default correlations the
DR probability density distribution becomes wider and bimodal. (Witt, 2014, p. 4273) is the first
to show such a bimodality feature. Then (Li, Shang, & Su, 2015) independently notice the same
fact when they observe that the frequency of the high and low risk categories (credit ratings) is
higher, than that of the mid-risk categories.
When the default correlation is negative, the DR distribution becomes narrow and goes
to a degenerate state to concentrate around the mean PD value. (Witt, 2014) does neither demon-
strate this, nor consider in his examples, though he accepts that negative correlation of Bernoulli
trials is feasible. In contrast to (Witt, 2014), (Van Der Geest, 2005) deals with the negative corre-
lation for the illustrative purposes. The illustration from the Figure 2 supports the statements by
(Preisser & Qaqish, 2014). They say that the variance of the correlated random variate distribu-
tions decreases for the low (negative) correlation and increases – for the high (positive) one.
15IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
60%
R=-0,07
50%
R=0
40%
R=0,1
Freq
30%
R=0,5
20%
10%
0%
0 1 2 3 4 5 6 7 8 9 10
# 1s
(a) PD = 20%
60%
50% R=-0,07
R=0
40%
R=0,1
Freq
30% R=0,5
20%
10%
0%
0 1 2 3 4 5 6 7 8 9 10
# 1s
(b) PD = 50%
Figure 2. Illustration of DR distributions for various PD and correlations of Bernoulli random variates.
Note: # 1s – the number of defaults (ones).
Compare Figure 2 to the statement by (Ozdemir & Miu, 2009, pp. 70-71, Fig. 2.4).
They claim that through-the-cycle (TTC) rating philosophy implies wider DR distribution,
whereas the point-in-time (PIT) rating philosophy leads to a narrow DR distribution. This data
simulation exercise vividly illustrates that (Ozdemir & Miu, 2009) claim is correct if and only if
they associate TTC rating philosophy with the higher default correlation and PIT one – with
lower one. However, we would expect an opposite, i.e. the DR distribution for the PIT DR is
wider than for the TTC one because of the volatility coming from the economic cycle stages.
Figure 3 depicts the dependence of the asset correlation values against the marginal PD
ones. Several lines correspond to the different values of DR variance. We advise the reader to
compare Figure 3 to Figure 1. As for the suggested approach illustration in Figure 3, we present
values from formula (3), i.e. the squared values ( ) for the visual comparability with Figure 1.
16IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
30
25
Asset Correlation (R), %
20 Var(DR)
1%
15 3%
10 5%
5 10%
0
0% 20% 40% 60% 80% 100%
Probability of Default (PD)
Figure 3. Illustration of the Suggested Asset Correlation (R) parametrization and its dependence with PD.
One may see that the regulatory asset correlation R formula (5), (6) in Figure 1 general-
ly reflects the concept of a declining asset correlation value for the higher PD values when PD is
less than 50%. However, the correlated Bernoulli random variates do not demonstrate the pat-
terns used in regulatory regulation for PD in excess of 50%. Then AC starts upward sloping (see
Figure 3). Such a positive dependence of asset correlation and default probability was underlined
by (Vozzella & Gabbi, 2010). Second, AC for the correlated discrete defaults varies because of
the changes in the DR variance. It cannot vary because of a credit product type given the same
PD as in Figure 1 unless this credit product is characterized by a different DR variance. This
means that the asset correlation can be uniquely statistically calibrated using the data on the
mean default rate and its variance.
As you can see from Figure 4, the suggested approach in formula (21) mostly ideally
predicts the DR variance values and quite closely fits the input correlation r value. The minor
discrepancies result from the fact that the optimized (iterated) correlation matrix for the generat-
ed joint distribution of Bernoulli variables does not always converge to the target correlation ma-
trix. One may note a high value of error, in excess of its limits when the algorithm reached the
maximum number of iterations, see column 7 in Annex 1.
We consider (Wunderer, 2019) as the most recent benchmark research for AC and DC
parametrization in Figure 4. He assumes the underlying correlated Gaussian asset returns when
estimating AC. See Annex 1, columns 9-11 for the DR variance comparison and columns 14-16
for the default correlation r comparison.
17IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
8%
7%
6%
5%
predicted
4%
3%
2% suggested
1% Wunderer
1:1
0%
0% 2% 4% 6% 8%
observed
(a) DR Variance
60%
estimate
40%
20%
0%
-20% -10% 0% 10% 20% 30%
-20% input
-40%
suggested
-60% Wunderer
1:1
-80%
(b) Correlation (r)
Figure 4. Comparison of suggested approach and that of (Wunderer, 2019) based on simulated data with
known characteristics.
The findings are listed below:
1. (Wunderer, 2019) overestimates the DR variance for the low (including negative) default
correlation and/or low PD. From the risk-management perspective that may be fine as the
approach produces the more conservative estimate of risk (proxied by DR variance, for
instance) than it really is;
2. (Wunderer, 2019) underestimates the DR variance for the high (positive) correlation
and/or high PD. This is not acceptable for the risk-management as the risk is underesti-
mated compared to its true value. Here the macroprudential add-ons should be used to
offset the credit risk underestimation.
18IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
3. As a consequence of the two above points, (Wunderer, 2019) overestimates the default
correlation r for the high PDs.
We conclude that (Wunderer, 2019) approach inadequately assesses the credit risk when
one has to deal with the correlated discrete default events. The reason is that (Wunderer, 2019)
assumes correlated Gaussian asset returns for the asset correlation parametrization.
5. APPLICATION TO REAL-WORLD: REGULATORY IMPACT
We use the publicly available data from the international credit rating agencies of
(Moody's, 2018, p. 27) and (S&P Global Ratings, 2019, p. 3). We compare the suggested ap-
proach in (24) assuming the correlated discrete defaults and the capital requirements assuming
the correlated Gaussian asset returns. We specifically estimate the mean DR and DR variance on
the two subsamples:
till 2003 inclusive (as if it had formed the basis for the (BCBS, 2004) AC functions’ cal-
ibration);
till 2018 (as if we updated the calibration using the recent data).
As the dominant part of the international credit rating agencies’ clients’ pool consists of
the non-retail (corporate) exposures, we assign the regulatory corporate exposure asset class
when evaluating capital requirements for them. Annex 3 provides tabulated results for the time
period covered by the data and the default rate mean and variance.
20%
correlated discrete defaults) to
capital requirements (assume
K1 / K0 - ratio of suggested
15%
alternative (assume…
10%
5% y = 9.5351x + 0.0176
R² = 0.9517
0%
0.0% 0.5% 1.0% 1.5% 2.0%
DR Coefficient of variation CV = Var_DR / mean_DR
Figure 5. Comparison of suggested and capital requirements with respect to the coefficient of variation (CV)
for the default rate.
Note: Horizontal axis has the coefficient of variance (CV). CV is the ratio of variance to the
mean value. (Miller, 1991) discusses its properties.
19IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
We find that the higher the default rate coefficient of variance (CV) is for the empirical
data under investigation, the closer the capital requirements from the suggested approach (cf.
(23) and (25)) assuming the correlated discrete defaults are to the IRB ones (see (4), (5), (6))
assuming correlated Gaussian asset returns. However, even in the extreme case, the latter are still
5 to 50 times higher than that originating from the suggested approach (see row 16 in Annex 3).
For instance, take S&P speculative grade (SG) borrowers’ default statistics for 1981-2018. The
mean DR is 3.9%, the DR variance equals to 0.07%. This implies the IRB capital requirement of
25.56% of the credit exposure (see row 10 of Annex 3) when assuming the correlated Gaussian
asset returns. If we consider the correlated discrete defaults (as the data actually is), it is 4.51%
(see row 14 of Annex 3). Thus, the ratio of the latter to the former equals to 4.51% / 25.56% =
17.7 %, i.e. the former approach assuming correlated Gaussian asset returns five times overesti-
mate losses than the latter that assumes correlated discrete defaults.
The capital requirements under the assumption of the correlated Gaussian asset returns
do not sufficiently well capture the discrete default data evolution from 2003 onwards. From one
side, for the speculative grades (SG) we observe a decrease in the mean DR and DR variance
disregarding the fact of the 2007-09 world financial crisis presence in the dataset. The IRB capi-
tal requirements for the correlated Gaussian asset returns do not fall as sharply as they do for the
correlated discrete defaults (compare rows 11 to 15 in Annex 3). We may accept this from the
risk-management point of view as the bank capital does not rapidly decrease either. However,
from another side, for the investment graded (IG) borrowers we observe an increase in the mean
DR and DR variance (e.g. for the Moody’s data). The IRB capital requirements assuming the
Gaussian asset returns in such a situation do not rise with the same pace (+20.5% from 2003 to
2018) as they do for the correlated discrete defaults (+28.2%). In contrast with the situation of
the speculative grades, this one is not acceptable from the risk-management perspective. Never-
theless, this may seem sufficient as the IRB approach assuming the correlated Gaussian asset re-
turns overestimates the credit risk amount in the absolute terms ca. 50 times for the IG grades
compared to the assumption of the correlated discrete defaults (see row 16 in Annex 3).
We wanted to verify whether the situation of the excessive IRB capital requirements as-
suming the correlated Gaussian asset returns takes place for any combination of the credit risk
parameters (mean DR and DR variance) compared to the values obtained for the correlated dis-
crete defaults. It turned out that this is not always true. For this purpose we ran the illustrative
calculations of the capital requirements for the different credit product types for the two values
of the mean DR (5% and 10%) and the two values of the DR variance (1% and 2%). Annex 2
presents the capital requirements in rows: No. 7 (for the regulatory approach assuming continu-
20IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
ous correlated Gaussian asset returns in formulas 3-5) and No. 10 (for the suggested
parametrization assuming correlated discrete defaults in formulas 22, 25). Row 11 of Annex 2
reports the ratio of the two capital requirements. When such a ratio exceeds 100%, we have the
situation of the credit risk underestimation according to the IRB capital requirements assuming
the continuous correlated Gaussian asset returns. So far we may see that when the DR coefficient
of variance (CV) is 40%, the IRB approach assuming the correlated continuous Gaussian asset
returns underestimates credit risk mostly twice compared to the one assuming the correlated dis-
crete defaults. The areas of the largest underestimation are plastic cards (QRR) and other retail
loans (see values of 239% and 209% in row 11 of Annex 2a, respectively). The revealed under-
estimation is substantial and requires implementation of the macroprudential add-ons to the IRB
risk-weights.
Same time, if CV=20%, the IRB capital requirements assuming the correlated Gaussian
asset returns overestimate the credit losses compared to the situation of the correlated discrete
defaults. Thus, the one has to compare an IRB credit risk estimate and the outcome resulting
from the actual discrete nature of the default events. When the credit risk underestimation is re-
vealed, macroprudential add-ons to the IRB risk-weights should be introduced to offset it.
6. CONCLUSION
Basel III (BCBS, 2017) and Basel Framework (BCBS, 2019) have inherited the IRB
approach from Basel II (BCBS, 2006). (Vasicek, 2002) model makes the foundation for the IRB
approach. It assumes normality of the borrowers’ asset returns. Vasicek assumes that returns are
correlated, i.e. there is an asset correlation. By introducing a threshold (quantile) for the Gaussi-
an distribution, he obtains Bernoulli trials. They reflect whether the default has occurred or not.
Due to the asset correlation, Vasicek arrives at the non-Gaussian unimodal portfolio loss proba-
bility density distribution.
Same time the IRB approach requires predicting the discrete events of the default. That
is why by construction the IRB approach deals with the Bernoulli trials. In such a case we may
call the correlation between the Bernoulli trials a default correlation. The correlated Bernoulli
trials were studied before (Witt, 2014; Zaigraev & Kaniovski, 2013; Chiu, Jackson, & Kreinin,
2017). The correlated Bernoulli trials have a bimodal default rate probability density distribution.
However, there was no an explicit parametrization for the default correlation assuming
discrete defaults before now. We are the first to offer such a parametrization. We show the im-
plications for the IRB credit risk assessment if the correlation of the discrete defaults was used
instead of the Gaussian asset return correlations. When the default rate coefficient of variance is
20%, the IRB capital requirements based on the correlated Gaussian asset returns exceed the
21IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
credit risk amount corresponding to correlated discrete defaults. When the default rate coefficient
of variance equals to 40%, the IRB capital requirements corresponding to the correlated Gaussi-
an asset returns are significantly lower than the amount standing for the correlated discrete de-
faults. To sum up, in case the regulator finds that IRB credit risk assessment is significantly un-
derestimated compared to the approach suggested in the working paper, the introduction of the
macroprudential add-ons to the risk-weights help to offset such an underestimation.
Notations used in Annex 1
- joint bivariate standard normal distribution function (see Annex 4);
iter., # - number of iterations used to obtain the output (limit of 10 thousand iterations
was set);
error – attained square root of sum of differences for two correlation matrixes: target and
actual;
AVar – analytically derived (predicted) variance;
indep – as if the correlation of Bernoulli variates equaled to nil;
column 14 equals to column 4; it is replicated for visible convenience to compare to col-
umns 15 and 16.
Suggested – approach from the current paper;
Wunderer – approach proposed by (Wunderer, 2019);
Notations used in Annex 2, 3
CV – coefficient of variation, ratio of DR variance to its mean value.
K – capital requirement to cover credit risk, equals to risk-weight (RW)
22IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
Annex 1. Simulated Data Analysis.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
# n PD r Ф2 iter., # error mean_DR Var_DR AVar_Corr AVar_Corr AVar Ф2 empir r_input r_empir r_empir
Suggested Wunderer indep Wunderer (=col. 4) suggested Wunderer
1 10 5% -10% 0,16% 10000 31,80% 5,40% 0,25% 0,05% 0,39% 0,48% 0,00% -10% -6% -60%
2 10 5% -5% 0,20% 10000 12,60% 5,40% 0,25% 0,26% 0,43% 0,48% 0,00% -5% -6% -60%
3 10 5% 0 0,25% 10000 38,00% 6,00% 0,38% 0,48% 0,48% 0,48% 0,16% 0% -4% -10%
4 10 5% 5% 0,31% 10000 53,90% 4,20% 0,39% 0,69% 0,53% 0,48% 0,16% 5% 0% -5%
5 10 5% 10% 0,37% 10000 80,10% 4,00% 0,63% 0,90% 0,58% 0,48% 0,43% 10% 7% 25%
6 10 5% 20% 0,52% 10000 86,00% 5,80% 1,42% 1,33% 0,72% 0,48% 1,30% 20% 18% 50%
7 10 10% -10% 0,72% 10000 22,40% 9,70% 0,13% 0,09% 0,65% 0,90% 0,11% -10% -9% -45%
8 10 10% -5% 0,85% 10000 23,60% 10,10% 0,45% 0,50% 0,77% 0,90% 0,52% -5% -6% -20%
9 10 10% 0 1,00% 10000 29,60% 9,50% 0,88% 0,90% 0,90% 0,90% 0,92% 0% 0% 0
10 10 10% 5% 1,16% 10000 33,00% 7,70% 1,11% 1,31% 1,04% 0,90% 1,03% 5% 6% 20%
11 10 10% 10% 1,33% 10000 28,40% 9,70% 1,63% 1,71% 1,20% 0,90% 1,77% 10% 10% 25%
12 10 10% 20% 1,72% 10000 61,30% 8,20% 1,83% 2,52% 1,55% 0,90% 1,86% 20% 16% 35%
13 10 25% -10% 5,26% 10000 18,40% 23,10% 0,40% 0,19% 0,98% 1,88% 3,80% -10% -9% -20%
14 10 25% -5% 5,75% 10000 12,50% 22,60% 1,00% 1,03% 1,43% 1,88% 4,28% -5% -5% -10%
15 10 25% 0 6,25% 10000 12,30% 24,10% 1,74% 1,88% 1,88% 1,88% 5,71% 0% -1% -5%
16 10 25% 5% 6,75% 10000 10,90% 26,80% 2,81% 2,72% 2,33% 1,88% 8,12% 5% 5% 10%
17 10 25% 10% 7,28% 8111 9,80% 27,30% 3,75% 3,56% 2,80% 1,88% 9,42% 10% 10% 15%
18 10 25% 20% 8,37% 10000 13,00% 23,60% 4,94% 5,25% 3,78% 1,88% 9,05% 20% 19% 35%
19 10 50% -10% 23,41% 10000 11,10% 47,90% 0,47% 0,25% 1,07% 2,50% 20,69% -10% -9% -10%
20 10 50% -5% 24,20% 4039 9,70% 48,90% 1,39% 1,38% 1,78% 2,50% 22,68% -5% -5% -5%
21 10 50% 0 25,00% 2592 9,40% 48,80% 2,55% 2,50% 2,50% 2,50% 23,87% 0% 0% 5%
22 10 50% 5% 25,80% 3766 9,40% 51,50% 3,62% 3,63% 3,22% 2,50% 27,77% 5% 5% 10%
23 10 50% 10% 26,59% 6128 10,20% 54,10% 4,67% 4,75% 3,93% 2,50% 31,70% 10% 10% 15%
24 10 50% 20% 28,20% 4933 9,10% 49,40% 7,01% 7,00% 5,38% 2,50% 29,41% 20% 20% 30%
23IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
Annex 2a. regulatory vs. suggested approach (illustrative case, mean DR=5%).
Data Descriptives
1 mean_DR 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0%
2 Var_DR 1,0% 1,0% 1,0% 1,0% 1,0% 1,0% 1,0% 2,0% 2,0% 2,0% 2,0% 2,0% 2,0% 2,0%
3 Exposure type Corporate SME HVCRE QRR Mortgage Other SIFI Corporate SME HVCRE QRR Mortgage Other SIFI
retail retail
4 row # (Annex 1) 1 2 3 4 5 6 7 1 2 3 4 5 6 7
Capital requirements assuming correlated Gaussian asset returns (K0)
5 r0 = sqrt (R0) 36,03% 29,98% 36,71% 20,00% 38,73% 22,93% 40,29% 36,03% 29,98% 36,71% 20,00% 38,73% 22,93% 40,29%
R_0 n/a -4% n/a 4% 15% n/a n/a n/a -4% n/a 4% 15% n/a n/a
R_min 12% 12% 12% n/a n/a 3% 15% 12% 12% 12% n/a n/a 3% 15%
R_max 24% 24% 30% n/a n/a 16% 30% 24% 24% 30% n/a n/a 16% 30%
coef -50 -50 -50 n/a n/a -35 -50 -50 -50 -50 n/a n/a -35 -50
6 R0 12,99% 8,99% 13,48% 4,00% 15,00% 5,26% 16,23% 12,99% 8,99% 13,48% 4,00% 15,00% 5,26% 16,23%
mean_DR 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0%
conf.level 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9%
7 K0 28,45% 22,57% 29,16% 14,73% 31,35% 16,81% 33,11% 28,45% 22,57% 29,16% 14,73% 31,35% 16,81% 33,11%
Capital requirements assuming correlated discrete default events (K1)
8 r1 21,05% 21,05% 21,05% 21,05% 21,05% 21,05% 21,05% 42,11% 42,11% 42,11% 42,11% 42,11% 42,11% 42,11%
9 R1 = r1 ^ 2 4,43% 4,43% 4,43% 4,43% 4,43% 4,43% 4,43% 17,73% 17,73% 17,73% 17,73% 17,73% 17,73% 17,73%
mean_DR 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0% 5,0%
conf.level 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9%
10 K1 15,46% 15,46% 15,46% 15,46% 15,46% 15,46% 15,46% 35,24% 35,24% 35,24% 35,24% 35,24% 35,24% 35,24%
Comparison of Capital Requirements
11 K1/K0 54,3% 68,5% 53,0% 104,9% 49,3% 92,0% 46,7% 123,9% 156,1% 120,8% 239,2% 112,4% 209,7% 106,4%
12 CV 20,0% 20,0% 20,0% 20,0% 20,0% 20,0% 20,0% 40,0% 40,0% 40,0% 40,0% 40,0% 40,0% 40,0%
24IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
Annex 2b. regulatory vs suggested approach (illustrative case, mean DR=10%).
Data Descriptives
1 mean_DR 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0%
2 Var_DR 1,0% 1,0% 1,0% 1,0% 1,0% 1,0% 1,0% 2,0% 2,0% 2,0% 2,0% 2,0% 2,0% 2,0%
3 Exposure type Corporate SME HVCRE QRR Mortgage Other SIFI Corporate SME HVCRE QRR Mortgage Other SIFI
retail retail
4 row # (Annex 1) 1 2 3 4 5 6 7 1 2 3 4 5 6 7
Capital requirements assuming correlated Gaussian asset returns (K0)
5 r0 = sqrt (R0) 34,76% 28,43% 34,82% 20,00% 38,73% 18,42% 38,86% 34,76% 28,43% 34,82% 20,00% 38,73% 18,42% 38,86%
R_0 n/a -4% n/a 4% 15% n/a n/a n/a -4% n/a 4% 15% n/a n/a
R_min 12% 12% 12% n/a n/a 3% 15% 12% 12% 12% n/a n/a 3% 15%
R_max 24% 24% 30% n/a n/a 16% 30% 24% 24% 30% n/a n/a 16% 30%
coef -50 -50 -50 n/a n/a -35 -50 -50 -50 -50 n/a n/a -35 -50
6 R0 12,08% 8,08% 12,12% 4,00% 15,00% 3,39% 15,10% 12,08% 8,08% 12,12% 4,00% 15,00% 3,39% 15,10%
mean_DR 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0%
conf.level 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9%
7 K0 41,24% 33,71% 41,32% 24,91% 46,34% 23,43% 46,51% 41,24% 33,71% 41,32% 24,91% 46,34% 23,43% 46,51%
Capital requirements assuming correlated discrete default events (K1)
8 r1 11,11% 11,11% 11,11% 11,11% 11,11% 11,11% 11,11% 22,22% 22,22% 22,22% 22,22% 22,22% 22,22% 22,22%
9 R1 = r1 ^ 2 1,23% 1,23% 1,23% 1,23% 1,23% 1,23% 1,23% 4,94% 4,94% 4,94% 4,94% 4,94% 4,94% 4,94%
mean_DR 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0%
conf.level 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9%
10 K1 17,26% 17,26% 17,26% 17,26% 17,26% 17,26% 17,26% 27,09% 27,09% 27,09% 27,09% 27,09% 27,09% 27,09%
Comparison of Capital Requirements
11 K1/K0 41,8% 51,2% 41,8% 69,3% 37,2% 73,7% 37,1% 65,7% 80,4% 65,6% 108,7% 58,5% 115,6% 58,2%
12 CV 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 10,0% 20,0% 20,0% 20,0% 20,0% 20,0% 20,0% 20,0%
25IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
Annex 3. regulatory vs suggested approach (empirical data).
# Moody's S&P Moody's S&P Moody's S&P
Data Descriptives
1 Grade Type Investement grade (IG) ALL (IG + SG) Speculative grade (SG)
2 Year Start 1983 1983 1981 1981 1983 1983 1981 1981 1983 1983 1981 1981
3 Year End 2003 2017 2003 2018 2003 2017 2003 2018 2003 2017 2003 2018
4 mean_DR 0,038% 0,049% 0,112% 0,091% 1,014% 0,963% 1,567% 1,443% 2,948% 2,566% 4,771% 3,994%
5 change after 2003 27,5% -19,0% -5,1% -7,9% -13,0% -16,3%
6 Var_DR 0,00005% 0,00010% 0,00011% 0,00014% 0,00574% 0,00587% 0,01031% 0,00969% 0,03829% 0,03811% 0,07437% 0,07158%
7 change after 2003 75,9% 22,6% 2,1% -6,0% -0,5% -3,8%
Capital requirements assuming correlated Gaussian asset returns (K0)
8 r0 = sqrt (R0) 48,76% 48,70% 48,32% 48,44% 43,85% 44,06% 41,81% 42,23% 38,40% 39,15% 36,20% 36,92%
R_0 n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a
R_min 12% 12% 12% 12% 12% 12% 12% 12% 12% 12% 12% 12%
R_max 24% 24% 24% 24% 24% 24% 24% 24% 24% 24% 24% 24%
coef -50 -50 -50 -50 -50 -50 -50 -50 -50 -50 -50 -50
9 R0 23,77% 23,71% 23,35% 23,47% 19,23% 19,41% 17,48% 17,83% 14,75% 15,33% 13,10% 13,63%
mean_DR 0,038% 0,049% 0,112% 0,091% 1,014% 0,963% 1,567% 1,443% 2,948% 2,566% 4,771% 3,994%
conf.level 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9%
10 K0 1,66% 2,00% 3,70% 3,18% 14,12% 13,78% 17,17% 16,57% 22,36% 21,09% 27,80% 25,56%
11 change after 2003 20,5% -14,1% -2,5% -3,5% -5,7% -8,1%
Capital requirements assuming correlated discrete default events (K1)
12 r1 0,14% 0,20% 0,10% 0,15% 0,57% 0,62% 0,67% 0,68% 1,34% 1,52% 1,64% 1,87%
13 R1 = r1 ^ 2 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,02% 0,02% 0,03% 0,03%
mean_DR 0,038% 0,049% 0,112% 0,091% 1,014% 0,963% 1,567% 1,443% 2,948% 2,566% 4,771% 3,994%
conf.level 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9% 99,9%
14 K1 0,04% 0,05% 0,11% 0,09% 1,06% 1,01% 1,65% 1,52% 3,23% 2,86% 5,29% 4,51%
15 change after 2003 28,2% -18,5% -4,7% -7,8% -11,6% -14,7%
Comparison of Capital Requirements
16 K1/K0 2,3% 2,5% 3,0% 2,9% 7,5% 7,4% 9,6% 9,2% 14,5% 13,6% 19,0% 17,7%
17 CV 0,1% 0,2% 0,1% 0,2% 0,6% 0,6% 0,7% 0,7% 1,3% 1,5% 1,6% 1,8%
26IRB ASSET AND DEFAULT CORRELATION: RATIONALE FOR MACROPPRUDENTIAL ADD-ONS TO RISK-WEIGHTS JULY 2020
Annex 4. Values of bivariate joint standard normal cumulative density function (Ф2 or where , ), in
percentage points (pp) out of 100.
R, %
-100 -75 -50 -25 -20 -15 -10 -5 0 +5 +10 +15 +20 +25 +50 +75 +100
1 0,1 0,3 1,0
5 0,1 0,1 0,1 0,2 0,2 0,3 0,3 0,4 0,4 0,5 0,6 1,2 2,2 5,0
10 0,1 0,4 0,5 0,6 0,7 0,9 1,0 1,2 1,3 1,5 1,7 1,9 3,2 5,1 10,0
20 0,1 0,8 2,2 2,5 2,9 3,2 3,6 4,0 4,4 4,8 5,2 5,7 6,1 8,7 12,1 20,0
25 0,3 1,8 3,9 4,3 4,8 5,3 5,8 6,3 6,8 7,3 7,8 8,4 8,9 12,0 15,9 25,0
30 0,9 3,3 6,1 6,6 7,2 7,8 8,4 9,0 9,6 10,2 10,9 11,5 12,1 15,7 20,0 30,0
PD, % 40 4,1 8,4 12,3 13,0 13,8 14,5 15,3 16,0 16,7 17,5 18,3 19,0 19,8 23,9 28,9 40,0
50 11,5 16,7 21,0 21,8 22,6 23,4 24,2 25,0 25,8 26,6 27,4 28,2 29,0 33,3 38,5 50,0
60 20,0 24,1 28,4 32,3 33,0 33,8 34,5 35,3 36,0 36,7 37,5 38,3 39,0 39,8 43,9 48,9 60,0
70 40,0 40,9 43,3 46,1 46,6 47,2 47,8 48,4 49,0 49,6 50,2 50,9 51,5 52,1 55,7 60,0 70,0
80 60,0 60,1 60,8 62,2 62,5 62,9 63,2 63,6 64,0 64,4 64,8 65,2 65,7 66,1 68,7 72,1 80,0
90 80,0 80,0 80,1 80,4 80,5 80,6 80,7 80,9 81,0 81,2 81,3 81,5 81,7 81,9 83,2 85,1 90,0
99 98,0 98,0 98,0 98,0 98,0 98,0 98,0 98,0 98,0 98,0 98,0 98,0 98,0 98,0 98,1 98,3 99,0
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